Priority-State Dynamics in Open Multi-Agent Systems: Dynamic Membership and History-Dependent Consensus Endpoints

Abstract

We characterize the mean priority returned by a single-agent arrival--mixing--departure history. Each row is a probability vector over the active objectives, so removing an agent also removes a coordinate and conditions the surviving rows. For arbitrary nonnegative row-stochastic mixing, we derive the returned mean directly from the incumbent state, retention masses and mixing matrix. Holding those inputs fixed gives the unique restoring newcomer conditional priority when its influence is positive, an exact strict-positivity feasibility test, and a separate zero-influence case. Under equal retention and doubly stochastic mixing, restoration requires a weighted average of incumbent rows; for one unweighted consensus update, this is the newcomer-neighbor row mean. The general result is an offline input-level characterization, not a distributed arrival protocol. An exact counterexample excludes unrestricted endpoint invariance. Standard final-epoch average consensus supplies the limiting-priority interpretation; no new physical-rendezvous or persistently open convergence theorem is claimed.